the total cost of producing x units of toys is C(x)=110-4x+〖.02x〗^2.
a.find the marginal cost
b. if 50 toys are manufactured, what is the rate of change of the total cost?
c.what is the amount in peso of the actual cost of manufacturing the 51st toy?
To find the marginal cost, we need to find the derivative of the total cost function with respect to the number of units produced, which is x.
a. Marginal cost is the rate at which the total cost changes as the number of units produced changes. In other words, it is the derivative of the total cost function.
Given that C(x) = 110 - 4x + 0.02x^2, we need to differentiate this function with respect to x to find the marginal cost.
C'(x) = -4 + 0.04x
Therefore, the marginal cost function is C'(x) = -4 + 0.04x.
b. To find the rate of change of the total cost when 50 toys are manufactured, we need to substitute x = 50 into the marginal cost function.
C'(x) = -4 + 0.04x
C'(50) = -4 + 0.04(50)
= -4 + 2
= -2
Therefore, the rate of change of the total cost when 50 toys are manufactured is -2.
c. To find the actual cost of manufacturing the 51st toy, we need to substitute x = 51 into the total cost function.
C(x) = 110 - 4x + 0.02x^2
C(51) = 110 - 4(51) + 0.02(51)^2
= 110 - 204 + 0.02(2601)
= 110 - 204 + 52.02
= -94 + 52.02
= -41.98
Therefore, the actual cost of manufacturing the 51st toy is -41.98 pesos.